0
votes
1answer
75 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
0
votes
0answers
24 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
0
votes
0answers
67 views

Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
0
votes
0answers
21 views

Using a theorem to find the center of a $p$-sylow subgroup of simple group

I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown ...
2
votes
2answers
70 views

Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
1
vote
1answer
29 views

regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
2
votes
1answer
41 views

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$

Let $G$ be a finite simple group. Suppose that $H$ is a subgroup of $G$ with index $n=|G:H|>1$. Show that $|H|$ divides $(n-1)!$ Hint: consider the action of $G$ on right cosets of $H$ in $G$. I'm ...
4
votes
2answers
79 views

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
2
votes
1answer
46 views

A question about the involution in simple groups.

Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question ...
0
votes
1answer
62 views

Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
1
vote
1answer
64 views

Size of maximal tori in finite simple groups of Lie type

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$? ...
2
votes
1answer
47 views

Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
0
votes
0answers
38 views

Existance of semisimple elements in a torus.

Let $G$ be a finite simple group of Lie type and $T$ be a maximal torus in $G$. Is it true that $T$ contains a regular semisimple element (a semisimple element which $C_G(x)=T$)? If yes, why?
5
votes
2answers
76 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
0
votes
0answers
19 views

dimension of a finie simple group of Lie type.

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$, namely $F_q$ of size $q$. Suppose $dim(G)=n$. What we can say about $|G|$? Is it true that $|G|=q^ns$, where ...
0
votes
1answer
75 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
0
votes
0answers
32 views

Brauer characters of finite simpl groups of type E8

I would like to know the Brauer characters of finite simple groups $E_8(2)$, $E_8(3)$ or $E_8(5)$. Is there any refrence for this topic? Thanks
4
votes
1answer
36 views

Automorphisms of spin groups over finite fields, even dimension

I don't know very much about spin groups, but I need to do some reasonably explicit hand calculations in the spin groups over finite fields of odd characteristic in even dimension (the Schur covers of ...
3
votes
1answer
87 views

Centralizer of unipotent element of simple groups

Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank. We know that ...
3
votes
2answers
136 views

Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy ...
7
votes
1answer
116 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
3
votes
0answers
64 views

Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
3
votes
2answers
142 views

Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
4
votes
2answers
163 views

An exercise about simple groups

Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...
4
votes
1answer
126 views

Prove that $G=S_{1}\times S_{2}\times\cdots\times S_{k} $

Let $G$ be a finite group and let $N_1,N_2,\ldots,N_k$ be normal subgroups. Suppose that $\bigcap_{i=1}^{k}N_{i}=\{1\}$ and that $G/N_i=S_i$ is simple. If all the groups $S_i$ are non-isomorphic ...
-1
votes
2answers
96 views

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. [closed]

Prove that if a finite solvable group is simple, it is a cyclic group of prime order. Help me some hints.
2
votes
2answers
88 views

Groups of order $56$

Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove? My idea: ...
2
votes
1answer
47 views

No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
7
votes
0answers
94 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
0
votes
1answer
81 views

Simple group with order $\geq n!$ cannot have subgroup of index $n$.

My problem is as seen in the title: For positive integer $n>1$, prove that a simple group with order $\geq n!$ cannot have subgroup of index $n$. Could anyone give me some hints on how to ...
6
votes
0answers
57 views

Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups

Can anybody tell me how to prove this theorem? Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p\in\pi(G)$ such that $G$ has no $\{2,p\}$-Hall ...
1
vote
1answer
54 views

non-split extension of the simple group $L_3(4)$

I would like to know the structure of the groups $L_3(4).C_2$ and $L_3(4).C_{11}$. (By $C_n$ I mean the cyclic group of order $n$ and by $G=K.L$ I mean the non-spli extension of $K$ by $L$, were $K$ ...
3
votes
1answer
61 views

Are quasinilpotent groups a Fitting class?

A finite group is called quasinilpotent if it induces inner automorphisms on all of its chief factors. A solvable group is quasinilpotent iff it centralizes all of its (necessarily abelian) chief ...
7
votes
1answer
149 views

Are there any distinct finite simple groups with the same order?

In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two ...
3
votes
0answers
153 views

The Monster group

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
0
votes
1answer
67 views

Projective linear group - solvable

Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
4
votes
1answer
162 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
10
votes
0answers
194 views

How confident can we be about the validity of the classification of finite simple groups?

The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed. How certain can we be that The proof of ...
4
votes
1answer
68 views

Sylow $p$-subgroups of $SO_3(\mathbb{F}_p)$

Let $G=SO_3(\mathbb{F}_p)$. We have $|G|=p(p^2-1)$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Is it true that $n_p=p+1$? We have the two conditions $n_p\equiv 1\mod p$ $n_p\mid ...
31
votes
2answers
458 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
30
votes
3answers
591 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
9
votes
1answer
440 views

Simple groups of order 168

How would I prove that there is at most one simple group of order 168? I've already seen that $GL_3(2)$ and $PSL_2(7)$ are simple groups of order 168, and I have seen direct proofs that they are ...
4
votes
1answer
178 views

On Group of order $30$ and $60$.

In this question on yahoo answers , the answer says , "with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ " my question is , how did " 6 * ( 5 - 1 ) " come from ? Which ...
4
votes
1answer
86 views

Is there a simple and a non-simple group with same numbers of elements of each order.

Are there finite groups $G$ and $H$ such that: $n:=|G|=|H|$. $G$ is simple. $H$ is not simple. for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ?
14
votes
1answer
336 views

The “architecture” of a finite group

I think that the aim of the finite group theory is the following: Given a generic finite group $G$, study completely the subgroup structure of $G$. There are at least two ways to achieve this ...
3
votes
2answers
301 views

Non-Abelian simple group of order $120$

Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
9
votes
1answer
351 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
2
votes
0answers
43 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
2
votes
2answers
780 views

Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple

Here's a question I've been asked; Given distinct primes $p,q,r$, show that any group $G$ of order $pqr$ is not simple. So far, my idea has been to individually check each possible proper subgroup, ...
1
vote
0answers
64 views

Soluble subgroups of finite classical simple groups

What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?